Given a measure space $(\Omega, \mathcal{A})$, let $\nu$ be a signed measure on that space and let $|\nu| := \nu^+ + \nu^-$ be the variation. Now consider the measurable functions $X:\Omega \rightarrow \mathbb{R}$ and collect all of them in a space we call $F$, i.e., $$F:= \{X:\Omega \rightarrow \mathbb{R}\}.$$ Let $V$ be the space of all finite signed measures on $(\Omega, \mathcal{A})$. Now consider the space $$K:=\{\nu\in V:\int_{\Omega}|X(\omega)|d|\nu|<\infty\}.$$
1.) Now, can there any intuition be gained about the signed measures in this space? E.g., what does it tell me to integrate with respect to the variation of a signed measure? Why not just integrate with respect to just the signed measure?
2.) Also it looks kind of similar to the definition of $L^p$ spaces. Is there any connection to them?
Next, consider the Hahn-Decomposition $\nu = \rho + \sigma$, such that $\rho$ is absolutely continuous to some measure $\mu$ on $(\Omega, \mathcal{A})$. That is, $\rho$ has a density w.r.t. $\mu$. Let us now define the space $$R:=\{\nu\in V:\int_{\Omega}|X(\omega)|d|\rho|<\infty\}.$$
I am reading some lecture notes where it is stated that now each $\nu \in R$ can be identified with the density of $\rho$ w.r.t. $\mu$.
3.) What exactly does that mean and why can we do that? Is there some bijection?
It is also stated that these densities of $\rho$ w.r.t. $\mu$ are in $L^q(\Omega, \mathcal{A}, \mu)$ if our $X$ are in $L^p(\Omega, \mathcal{A}, \mu)$. Furthermore, it is being said that the pairing on $F\times V$, where $V$ is the vector space of all finite signed measures on $(\Omega, \mathcal{A})$, given by $\langle X, \nu \rangle=\int X dv$, this pairing comes down to the usual pairing between $L^p$ and $L^q$, i.e., for $f\in L^p$ and $g \in L^q$ we have $\langle f, g \rangle=\int f g d\mu$.
4.) Is there an obvious way to see that?
5.) Does that mean that the space $R$ is the dual space of $L^p$?
Let me try to answer your questions.
I think the best way to think about $K$ is as the space of those signed measures $\nu$ which can be made into linear functionals on $F$ as $$\label{1} X \mapsto \int_\Omega X \, d\nu. \tag{1}$$ If you think about it this way, then it is clear why we need to define $K$ the way it is as we need $$\int_\Omega |X| \, d|\nu| < \infty$$ to make sense of the integral in \eqref{1}.
One of the connections to $L^p$-spaces is the example given in your lecture notes (which I gather your remaining questions are about).
Reading your lecture notes I gather that in the remaining questions we set $F = L^p(\Omega, \mathcal{A}, \mu)$ for some fixed measure $\mu$ on $(\Omega, \mathcal{A})$.
I guess the point of introducing $R$ here is that $K$ is quite unnatural in this case. The reason being my answer to question 1 that we should think of $K$ as a space of linear functionals on $F$. The problem here is that $L^p$ is not a space of measurable functions but a space of equivalence classes of measurable functions, where two functions are equivalent if they agree on a $\mu$-null set. Hence if $\nu$ is a measure which is singular to $\mu$, then there is no way to make sense of \eqref{1}. I therefore think that $R$ should be defined a bit differently, namely as those elements of $K$ which are absolutely continuous with respect to $\mu$. It is then clear how to identify $R$ with the space of densities by using the Radon-Nikodym theorem, and it is clear that $R$ is the subspace of $K$ which are well-defined linear functionals on $L^p$.
To show that the density of $\rho$ is in $L^q$ for $\frac{1}{p} + \frac{1}{q} = 1$ you can use the same techniques as one uses to show that the dual of $L^p$ is $L^q$. For the second part, if $M$ is the density of $\rho$, then for any $X \in L^p$ we have $$\int_\Omega X \, d\rho = \int_\Omega XM \, d\mu = \langle X, M \rangle$$ by the Radon-Nikodym theorem.
$R$ is not necessarily the dual of $L^p$ since we only consider finite signed measures.